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AP DSC School Assistant Math Model Test 3

Section: Subject Specific

Question : 104 of 160

Marks: +1, -0

What is the value of tan2 165° + cot2 165° ?

Solution:

Concept:

Use trigonometric identities for angles in the second quadrant and known exact values for tan 15° and cot 15°.

Explanation:

Given expression:

tan^{2}165° + cot^{2}165°

Step 1: Express 165° as 180° − 15°.

tan 165° = tan(180° - 15°) = -tan 15°

cot 165° = cot(180° - 15°) = -cot 15°

Step 2: Square both to eliminate the negative sign.

tan^{2}165° = tan^{2}15°

cot^{2}165° = cot^{2}15°

Step 3: Use known values:

tan 15° = 2 - \sqrt{3}

, so

tan^{2}15° = (2 - \sqrt{3})^{2} = 4 + 3 - 4\sqrt{3} = 7 - 4\sqrt{3}

cot 15° = 2 + \sqrt{3}

, so

cot^{2}15° = (2 + \sqrt{3})^{2} = 4 + 3 + 4\sqrt{3} = 7 + 4\sqrt{3}

Step 4: Add the two results.

(7 - 4\sqrt{3}) + (7 + 4\sqrt{3}) = 14

Answer:

The value is 14, which corresponds to option B.

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